An equation of the tangent plane:
(
x
 2) + 2(
y
 (4))  2(
z
 5) = 0.
You may obtain an equivalent standard form varmint if you wish.
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16. (5 pts.)
The equation
r
4cos(
θ
)
10sin(
θ
)
is that of a cylinder in cylindrical coordinates.
Obtain an equivalent
equation in terms of rectangular coordinates (x,y,z).
Provide a vector
equation for the straight line that is the axis of symmetry.
Multiply the given equation by
r
.
Then doing the usual conversion and
completing the square a couple of times yields
(
x
2)
2
(
y
5)
2
29.
A vector equation for the line of symmetry in 3space is
<
x
,
y
,
z
>
<2,
5,
t
>,
for t
∈
.
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17. (5 pts.) The point (3,4,5) is in rectangular coordinates.
Convert
this to spherical coordinates (
ρ
,
θ
,
φ
).
[Inverse trig fun?]
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18. (5 pts.) Do the lines defined by the equations
<x,y,z> = <0,1,2> + t<4,2,2> and <x,y,z> = <1,1,1> + t<1,1,4>
intersect?
Justify your answer, for
yes
or
no
does not suffice.
If the lines are to intersect, there must be numbers t
1
and t
2
so
4t
1
= 1 + t
2
, 1  2t
1
= 1  t
2
, and 2 + 2t
1
= 1 + 4t
2
.
[Here t
1
is the
parameter for the putative point in terms of the first equation and t
2
is
the parameter value for the second equation.]
Solving this system yields
t
1
= 1/2 and t
2
= 1.
Using either t
1
or t
2
in the appropriate vector
equation yields the point of intersection, (2, 0, 3).
Note, however, you
really do not have to obtain the point itself to answer this question!!
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19. (5 pts.) What is the area of the triangle in three space with vertices
at P = (3, 0, 0), Q = (0, 4, 0), and R = (0 , 0, 4).
Let
v
be the vector with initial point P and terminal point Q, and let
w
be the vector with initial point P and terminal point R. Then the area A
of the triangle is given by
A =
v
×
w
/2 =
<3,4,0> × <3,0,4> /2
=
<16,12,12> /2 = 4 <4,3,3> /2 = 2 <4,3,3>
= 2(34)
1/2
= (544)
1/2
/2 ???? regressing.
..???
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20. (5 pts.) Do the three 2space sketches of the traces in each of the
coordinate planes of the surface defined by
.
z
1
x
2
9
y
2
4
Do not attempt to do a 3  space sketch.
If you don’t have enough
space below, say where any additional work is.
The xyplane and the xzplane appear below.
The yzplane appears on
Page 2 of 5.
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 Spring '06
 GRANTCHAROV
 Multivariable Calculus, pts, Euclidean geometry

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